| Statistical Independence |
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| CATEGORIES ABOUT STATISTICAL INDEPENDENCE | |
| probability theory | |
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Similarly, when we assert that two Random Variable s are independent, we intuitively mean that knowing something about the value of one of them does not yield any information about the value of the other. For example, the number appearing on the upward face of a die the first time it is thrown and that appearing the second time are independent. INDEPENDENT EVENTS The standard definition says: :Two events ''A'' and ''B'' are independent Iff P(''A'' ∩ ''B'') = P(''A'')P(''B''). Here ''A'' ∩ ''B'' is the Intersection of ''A'' and ''B'', that is, it is the event that both events ''A'' and ''B'' occur. More generally, any collection of events -- possibly more than just two of them -- are mutually independent iff for any finite subset ''A''1, ..., ''A''''n'' of the collection we have : This is called the ''multiplication rule'' for independent events. If two events ''A'' and ''B'' are independent, then the Conditional Probability of ''A'' given ''B'' is the same as the "unconditional" (or "marginal") probability of ''A'', that is, : There are at least two reasons why this statement is not taken to be the definition of independence: (1) the two events ''A'' and ''B'' do not play symmetrical roles in this statement, and (2) problems arise with this statement when events of probability 0 are involved. | ||
| : P(''X'' | ''x'', ''Y'' = ''y'' ''Z'' = ''z'') = P(''X'' = ''x'' ''Z'' = ''z'') · P(''Y'' = ''y'' ''Z'' = ''z'') |
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| : ''p''<sub>''XY''''Z''</sub>(''x'', ''y'' ''z'') | ''p''<sub>''X''''Z''</sub>(''x'' ''z'') · ''p''<sub>''Y''''Z''</sub>(''y'' ''z'') |
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| : P(''X'' | ''x'' ''Y'' = ''y'', ''Z'' = ''z'') = P(''X'' = ''x'' ''Z'' = ''z'') |
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